#238884

order	512
nilpotency class	3
isoclinism	isoclinic class of #238875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1007 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11)(4,7)(5,9)(6,10)(8,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,26)(23,27)(24,31)(28,30)(29,32);(2,7)(3,12)(4,5)(8,15)(9,10)(13,14)(17,19)(18,20)(21,28)(22,24)(23,29)(25,30)(26,31)(27,32);(1,3,4,14,6,13,5,12)(2,8,9,16,10,15,7,11)(17,20)(18,24)(19,25)(21,30)(22,31)(23,32)(26,29)(27,28);(3,13)(8,15)(11,16)(12,14)(17,21,23,22)(18,25,27,26)(19,28,29,24)(20,30,32,31)

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146080);